Can I get the same quality of business card print if I decide to do it using my own printer?

Th professional printer will use offset for business cards. You will only be able to print on digital or laser print, more expensive in the long run.
Home printing will give a good quality card, offset will give a more expensive feel.

My business cards have raised printing (the ink has a thickness to it that you can feel) on very good quality stock, perfectly aligned and crisply cut. You can’t get all that with your own printer. So it depends on how much quality you mean by quality. You can do a decent job on your printer, but not great.

I have my cards done by Iconix.

No, you cannot, but you can get a fine looking product.

No but you can get special printing paper and card, with this and some printing fine tuning you can get a good result.