Square numbers which end in 1 or 9 always have an even number in the 10s column. (01, 09, 49, 81, 121, 169...) Square number which end in 6 always have an odd number in the 10s column. (16, 36, 196, 256...) Square numbers which end in 5 always in fact end with 25, so never with 55.
Square numbers which end in 4, however, always have an even number in the 10s column (04, 64) and so we get plenty ending in ..-44 (starting with 144 which is 12 squared) and even ...444 (starting with 1444 which is 38 squared). But there it stops. If the last three digits are 4's, the number in the thousands column is odd: 462 squared is 213444, 538 squared is 289444, etc.
What about other bases? Well, in binary, every even number's square ends with at least two zeros (because they are all multiples of four), and every odd number's square ends in ...001 (because they are always one more than a multiple of eight).
In hexadecimal we start off promisingly:
529 (23 squared) in base 10 is 211 in hex
2116 (46 squared) in base 10 is 844 in hex
3481 (59 squared) in base 10 is E99 in hex
4761 (69 squared) in base 10 is 1299 in hex
6724 (82 squared) in base 10 is 1A44 in hex
11025 (105 squared) in base 10 is 2B11 in hex
12100 (110 squared) in base 10 is 2F44 in hex
91204 (302 squared) is 16444 in hex
521284 (722 squared) is 7F444 in hex
762129 (873 squared) is BA111 in hex
Now we have to go up considerably in scale, and there aren't any squares ending in more than five repeated 4's, as the sixth last digit is always odd (1's and 9's, however, continue)
48209289910681 (6943291 squared) in base 10 is 2BD899999999 in hex (I haven't found any smaller squares ending with seven repeated digits)
And I guess that we can keep going with steadily increasing numbers of 1's and 9's (this is probably trivial enough to prove).
Counting in an odd-numbered base, it is also pretty easy to build up repeated digits as the distinction between odd and even numbers no longer applies.
121 (11 squared) expressed in base 3 is 11111
Just thought you ought to know.