Squares with No More than 2 Distinct Digits

Sequence
The sequence of square numbers which have no more than $2$ distinct digits and not ending in $0$ begins:
 * $1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 225, 441, 484, 676, 1444, 7744, 11 \, 881, 29 \, 929, 44 \, 944, 55 \, 225, 69 \, 696, 9 \, 696 \, 996, 6 \, 661 \, 661 \, 161, \ldots$

It is not known whether there are any more.