This is not possible. The code automatically selects the closest corner to the end of the cut (or origin if nothing cut yet) for the start of the next cut. This is a vast improvement on accepting sketchups default line draw order, however there is no way to select a corner and tag it as 'start here'.

If things are flying apart you need to use tabs to hold them together (-:

You can influence the cut order by grouping items (plus their cut lines, so do all cut lines before grouping), things will be cut in group order.
There is a group reorder tool on the toolbar.
There is a group order visualizer on the Tools|Phlatboyz menu (name the groups to make this useful)
You can embed groups inside groups.
Groups will cut before all ungrouped cuts.

So in your example above I would group the three beams, then make the middle one cut first, this ensures that the top and bottom ones are still firmly attached until they are cut. But really, tabs are for holding cut stuff together, either use them or use more material or set cut the order with groups.

 

an alternative to tabs is to use 'onion skinning'.
say you are cutting 6mm material that you could cut in one pass, but if you do that the piece comes loose and causes trouble.
so, instead of doing a single 100% deep pass, set multipass on and set the depth to 5.5mm.
this will now do 2 passes, doing the first one 5.5mm deep and leaving a 0.5mm 'onion skin' behind, which holds the part in place.
the next pass will cut through the skin with very little force, leaving a clean edge.

you can of course do this with multipass, just calculate the pass depths to leave a skin. so for 6mm plywood that you do in 2mm deep passes, 3 passes gets you to 100%, but to leave a 0.5mm skin set multipass to 1.84mm (5.5/3, rounded up).

if you are using a vacuum table then onion skinning can really help with holding everything down, since every cut through the material allows vacuum to leak. to do this, set your material thickness to 5.5mm (for 6mm material) and generate your main code, using multipass if needed. this will leave a 0.5mm skin. now set material thickness to 6mm, set multipass off, and generate another gcode file. join them together and cut. you can even increase the feed speed for the final cut since it is only cutting 0.5mm.

another alternative is to tab as usual (with or without multipass) and then do a 100+% deep single pass to clear the tabs.
to do this you need to create a copy of the drawing. on copy 1 you do tabs etc, on copy 2 you leave off the tabs.
for copy 1, set your multipass parameters as usual and generate gcodefile1
on copy 2, set parameters for NO multipass and generate gcodefile2
now use the Gcode joiner to join them together, so the main cutting is done with tabs and then a final 'cut off the tabs' pass is done.

Please watch these videos for more details on doing multi part drawings...

 

how are you holding it down? if you vacuum it down it will be as flat as the bed. making a vacuum table is quite easy. 12mm MDF with grooves in it, some paint to seal it on the edges, and it will even hold itself down since MDF is porous and will actually suck on the bottom side without grooves.

they produce the same sort of cut but the fold tool automatically shorten the line so it does not intersect with the outline of the part.
to quote the help ( in the toolbar)

Code:

Use the ["End" key] to toggle between short and wide mode. When hovering over an edge, the short mode shows a pink color preview. Wide mode shows a darker red/purple preview color. Short and Wide mode status is also shown on the bottom status text. The default short mode will shorten both ends of the edge by a small amount. The main reason is to break contact and stop the possible creation of an extra face and loop, which could confuse SketchUcam. Wide mode will act normally and not offer this protection. But you can use wide mode, if say you want a connected chain of edges. 

Some years back when I started working on SketchUcam I added the cut optimizer which reduces air time between cuts. This automatically finds the closest point on the next cut to start from. What you are seeing is the side effect of this process, by rotating the part different points are found as the closest point from the last cut position. It will not start the cut on really short segments either, if ramping is on, because we cannot ramp short segments effectively.

Cut order is affected by drawing and editing order (and seemingly random internal stuff in Sketchup) , unless you group the parts, in which case they will be cut in group order, which is why there is a group re-order tool and also a 'groups summary' viewer that displays the groups in cut order. groups needs to have names for this to be useful!

Now, this knowledge allows us to mess with the system (just had this idea!). By placing a very shallow cut, 1% deep , near a corner where we want cutting to start, and offsetting Z so a 1% deep cut does not touch the surface , and grouping in the right order, we can force the shallow 'missed cut' to be just before the corner we want to start at, thus forcing a start point.

yep, a quick test shows it will work. I placed 2 squares with outside cuts. placed a plunge hole at top right of the left hand one, and bottom right of the right hand one. the plunge holes are 1% deep. material is 6mm, overcut% is 105% which means it will cut to Z-6.3
1% is Z-0.060. so, zero Z on the surface, then raise it 0.06mm and re-zero. set the group order, lefthole,leftsquare,righthole,rightsquare and the job is done. if the fake holes take time you cannot afford, search and replace Z-0.060 with Z0.5 and those lines will do nothing (assuming you are running V1.4 in metric mode)

btw you can also affect cut start points by placing a tab on a cutline and then removing it. This leaves behind split points in the cut line which are detected as though they are corners by the optimizer, and if one of those points is closest to the previous cuts end, cutting will start there. This same technique is used for adjusting ramping and IS mentioned in the help. If you have not read the help(yes, all of it , then there is a bunch of stuff you don't know that could be useful to you (and you won't have to wait a day for me to answer your questions).