Ed, Assuming that you're going to put your piece outside in the weather, you'll need an adhesive that will pass a Type I test. Breifly the specifics of that test are that a 1x1 piece will pass a 3 cycle water boil/dry test without significant delamination. The key there being significant, a small amount of shallow delamination at the joint is allowed. Here's your options:
1- Titebond III - rated to pass type I, probably doesn't in all cases and applications (Franklin's results would have been carried out under perfect lab conditions). Not terribly expensive, safe to use, glueline is almost invisible, easy to find.
2- PRF or RF resin - passes type I EASILY, is expensive, difficult to find, has relatively hazardous chemicals (although I wouldn't be afraid to handle myself). Requires good clamp pressure and tight fitting joints. Needs at least 70 degree temp during cure, which is at least overnight. Also gives BLACK gluelines, which will be noticilbe in your cypress, they'll look like the glue joints in exterior grade plywood.
3- PUR (Polyurethane, or Gorrilla Glue) - passes type I easily, moderately expensive, relatively hazardouls chemical (again I would use it in my basement shop). Fairly easy to find. Requires good clamp pressure and tight fitting joints, I cannot stress this enough. PUR's do not act as gap fillers! Will give you a nice glueline that will look nice with your cypress. If I were building what you're taking about, this is the way I would go. You'll want to spritz the wood surfaces with a small amount of watern first, here's how I use it at home. Get an old windex type sprayer, spritz both wood surfaces, spread the adhesive, and clamp over night. There is a foaming action as the adhesive cures, it is there to drive the adhesive into the wood pores, the action is very stong, hence the need for an unusual number of clamps. If you don't have access to as many clamps as Norm Abrams has, don't even this about this, a few clamps is NOT going to get the job done.
4- You could also consider epoxy, but if you go this way, get out your checkbook.
Regarding your question on UF resin, In a cold press application, like you have at home, the best your should be able to get is a type II bond. This is a less strigent test than the type I above. It's for indoor applications or very limited outdoor exposures only.
Hope this helps, PM me if you need more details.