Determining whether a system is LTI given response to a rectangle box input
With just the kind of information you have, you can't prove the system is LTI. (As Dave points out in his answer, you can definitively prove this particular system is not LTI)

Specifically, for a system with input \$x(t)\$ and output \$y(t)\$, for the system to be called time-invariant, you need to know that if you time-shift the input by \$t'\$ so that you have \$x(t-t')\$, then the output is \$y(t-t')\$. In your case you only know the output for the case \$t' = 0\$, and you haven't been told that the input-output relationship will be maintained if the input signal is advanced or delayed in time.

Generally, time-invariance is something you have to be told about a system, or you have to assume it based on the physical nature of the system. Because if it's possible for the system to not be time-invariant, then any observations you make on it don't prove the system behavior won't change at some later time.