I would be much more concerned about things like weld quality than round vs. square.
With typical box tubing and round tubing, the material is typically structural steel with relatively low properties (not saying you can't get some with elevated properties). In the case with structural shapes, unless the tube is pre heat treated, you are working with a steel in the annealed state. That is what the localized area around a weld will be. So in this case you are probably not deteriorating the material properties. That being said, with different materials, different things can happen during the welding process depending on material and process used, inert gas, etc... It is a very broad field that requires a fair amount of knowledge.
If you are interested in some absolutely fantastic reading that is a bit old school but incredibly informative material on welding (at an extremely low price!) I would recommend you look at purchasing:
https://ssl.lincolnelectric.com/lincoln/apdirect/item.asp?prodnum=DWS
https://ssl.lincolnelectric.com/lincoln/apdirect/item.asp?prodnum=TS
I have a few of Blodgett's books and they are the best I have seen on welding. I can't believe how inexpensive they are for the amount of information they provide.
BTW...nice bit of work there. Congrats!