In this cross-number two-digit numbers are prime numbers, three-digit numbers are square numbers and four-digit numbers are palindromic numbers.
441 144 841 484 411 441 379 119 289 196 441 361
Reasoning for the top right corner:
The 2-digit prime starting with 4 can be 41, 43 or 47; since no squares end with 3 or 7, it's 41.
Next, the right column has to be a square ending with 1 consisting of digits that a square number can end with: 0, 1, 4, 5, 6 and 9. The two candidates are 441 and 961, and 961 doesn't fit because of the top row: no square ending with 9 has 2nd digit which would be a start of a square ending with 4. Thus the right column is 441.
Finally, to fill the rest there are squares 144, 324, 484 and 784 ending with 4, of which 324 and 784 can't be used in rows, since either 3 or 7 would have to be the last digit of a square (due to the 4-digit palindromic numbers) or 2 or 8 would have to be the first digit of a square ending with 4. This finishes the square.
For the rest of the digits the logic is pretty much the same, so I'll skip it.