You know when you enlarge a picture really big, you can see the squares (pixels?)? Is there a program that kind of fills in the gaps or redefines it so the picture can be enlarged without looking bad?

No, You can never get more detail than was originally captured. You can scale the picture up or re-sample it but neither will give you more detail.

simply put, scaling will make the print look pixelated.
Re-sampling will make the picture look like it has mutated.

The only way would be to re-scan the image at a higher resolution.