The issue is not whether you invest in the same fund in two accounts. The issue is the total amount of money you have invested in that fund. In other words, if you have $10,000 of money invested, and you don't want it all invested in the same thing, then don't invest it all in VFINX. It doesn't matter whether you invest some of it in VFINX in your 401k and some of it in VFINX in another account; that's still investing all of it in VFINX. If, on the other hand, you do want to invest everything in VFINX, then go ahead. The point is that it, for purposes of having "too many eggs in one basket", it doesn't matter very much what accounts you're holding the investment in; what matters is what actual financial instruments (e.g., which stocks, which mutual funds, which ETFs) you buy.
It seems your real question, then, is "is it a good idea to invest all my money in one mutual fund?" The answer is likely no, but there are certainly people who do that (and there are certainly people for whom VFINX is that one fund). It is certainly reasonable to want to diversify your holdings into a few different funds. (But again, what matters is what funds you buy, not which account you buy them in.)
You may want to read up on asset allocation. If you follow an index-fund-based strategy, the general idea is that you would want to identify several broad asset classes (e.g., US stocks, international stocks, corporate bonds) and then invest in a small number of mutual funds targeting those asset classes. If that is the way you want to do things, you might want to check out the Bogleheads asset allocation page, which explains asset allocation from the perspective of the "Bogleheads" philosophy (which involves, among other things, always investing in cheap index funds). Which asset allocation is right for you is a personal choice that depends on your age, anticipated retirement expenses, psychological makeup, and other factors, but reading the materials there should get you started. If you do get to the point where you need to consider how to distribute your asset allocation across multiple accounts, there is a Bogleheads page on that as well.
(I should note that, for tax purposes, it often does make a difference which account you hold investments in, but that is a separate question and doesn't seem to be the focus of your question here.)