If your project in Illustrator was made of vectors, you shouldn't have issues with the resolution. Vectors are the most flexible and quality kind of files, and can be printed at any size.
By default, vectors have a transparent background even if you get a message about flattening the content.
But if you used transparencies, it's possible indeed that you'll need to slightly modify your artwork. It's hard to tell you what exactly should be done without seeing your vector and without knowing if you placed rasterized images into it. The type of t-shirt printing process you'll use might also require you to flatten transparency and the blending effects, and send a perfect vector-only file.
When opening that EPS in Photoshop, the quality should be very good if you open it at 300ppi and more; the difference you might see is if you zoom in in Photoshop, you'll see pixels and not perfect lines. It's normal because Photoshop destroys that vector information and convert your image to pixels. To know if your image is alright, there's no need to zoom in at more than 150%; what-you-see-is-what-you-get and at 100-150%, if it looks good on your screen, it should look good when printed too.
Your printer will probably use the vector in a vector software such as Illustrator, and the quality will be as good as when you open your own file in Illustrator. Just so you know, Photoshop and Illustrator offer the EPS format but they're still different. The one from Illustrator can keep its transparency (eg. no white background) and vector information, while Photoshop will really flatten the artwork and remove the transparency (eg. add a white background), unless you use a clipping path.
When you save as EPS, you should also make sure you selected the "high resolution" in the transparency preset, instead of medium resolution.
The file you'll send to your printer is the EPS from Illustrator;
don't send a EPS opened in Photoshop. You can open your EPS in Photoshop to verify the quality but it's the EPS from Illustrator that you should use for print.
What you can do to make the vector transparency in "real" vector without rasterizing it:
For some projects, it's ok to simply select the elements that have special effects or transparency and use the "rasterize" in Illustrator to merge these effects together. But it won't be vector anymore, and there might be some white backgrounds on some parts of your design. For your t-shirt project, I'm not sure if it's something you want.
So one thing that can be done is to use the "pathfinder tool" and break (divide) your artwork in smaller pieces. Then you'll make sure the opacity and blending is set to 100% and normal on all these parts and... you'll fill these new parts back with the color they should be. In other words, you will imitate that effect of transparency and blending by using solid colors this time!
The easy way to do this is by using as a reference a rasterized image (eg. JPG or PSD) of your artwork before you used pathfinder on your artwork, and use the color picker to get the color recipe of each parts of your design. And then select each part in Illustrator and apply that color to it. Technically, if you didn't add any raster image in your Illustrator, you should end up with a vector that has no transparency and the final result should look the same or very very close to what you did originally. You probably won't get any "transparency flattening" message once you save that file in EPS.
That might be a bit more tricky if you used spot/pantones colors but at the same time, if having spot colors is important then it's not transparency or blending effects you should have use; you should use overprint attribute (eg. trapping) and different density of that Pantone to create a real print-ready. In doubt, simply ask your printer if the color separation is important (eg. if he needs to have each color in "spot" or if the print will be in 1-2-3 colors instead of "process colors")
Finally, if to your knowledge you didn't use any transparency or blending effect but still get that message, it's possible one tiny part of your document uses a different opacity or blending. You'll need to find it.
Don't forget to keep a duplicate of your original file with the transparency before doing this! It will be easier to make changes later if you need to and the technique above is very "destructive"!