Proving that sequences with the pattern aaaaaaa don't include a perfect square

No square ends in $3$ or in $7$.

No square ends in $11$.

But some squares end in $44$, but never mind if $44\cdots 44$ is a square, so is $11\cdots11$ which isn't.

For 1 you are done. For 2 not divisible by 4. For 3 no perfect square ends with digit 3. For 4 just divide by 4 and get back to the case with 1. For 5 not divisible by 25. For 6 not divisible by 4. For 7 no perfect square ends in digit 7. For 8 not divisible by 16. For 9 divide by 9 and get back to 1.