There are three issues with compatibility here.
- Mechanical and electronic
- Field of View
- Optical Quality
Mechanical and electronic
The EOS DSLR mounts should be mechanically compatible with those lenses as they have EOS mounts.
However the mount electronics have developed over the years and it is possible, as another poster said, that they will not work correctly. It's a try-and-see thing, unfortunately.
Sigma, AFAIK, do sometimes offer a service to upgrade the ROMs in old lenses to make them compatible with modern DSLRs. I do not think Tamron do that at all. Worth emailing Sigma to find out anyway.
Field Of View
The sensor on a Canon 600D is smaller than the standard frame of 35mm film used in film EOS cameras. This means that you actually see a crop of the image cast by the lens over what you would see in a film camera. That crop-effect is equivalent to have a lens with a focal length of about 1.6x the stated focal length.
This is why crop frame DSLRs, like the 600D, have kit lenses that are 18-55mm in focal length. When you allow for the crop-factor, this gives an approximately equivalent field of view to that of a 29-88mm lens would on a *full frame** ( standard 35mm film frame ) camera, like your EOS film cameras.
So while the lens might work, the effective field of view it gives you is not going to be what you were used to. This may work out better for you ( if you tend to use longer focal lengths anyway ), but may not.
Modern lenses, even kit lenses, are, generally, a step up from most older zoom lenses. You cannot be absolute about this, but as a rough rule-of-thumb that' how it works.
So a modern 18-200mm lens ( which on a 600D would have a similar field of view to your 28-300mm on a film camera ) might be better optically in some ways ( but might not - again it's a try-and-see thing ).
The kit 18-55 is generally very good optically compared to older kit lenses, BTW.
Modern coatings have also improved, and that can be a significant issue in reducing flaring.
It is something to consider, especially as you can often find used lenses from dealers at reasonable prices.