3 weeks ago I noticed that the caulking around the base of the toilet looked "rusty", I told the maintenance man about it and that I thought the wax seal probably needed to be replaced. He looked at it and his solution was to replace the rusty caulking with new caulking. 2 weeks ago, I notice that when I flushed the toilet, I would hear gurgling from the bathtub drain. Then after the 4th or 5th flush of the toilet that day, sewage backed-up into the bathtub and under the toilet mostly from the toilet bolts. Did the wax seal cause this or is it something else?

No, a blocked sewer pipe caused the back up.
If he only caulked around the pipe he did nothing but hide the problem. The old caulking was probably contaminated by the toilet backing up. If he replace the wax ring he could have gotten some of the wax ring in the sewer pipe and caused the blockage.
Doesn't matter, his problem, call him back.