I have partial ownership of a private C Corporation. The list value of
the common stock is 1 cent per share when it was first registered. If
the corporation were to sell today (tax issues aside) the stock would
be worth around $2 per share based on the average selling price for
similar corporations in the industry.

Would this stock be acceptable as collateral for a bank loan?

Probably not, I think.

However, if you were getting some other type of loan such as a real
estate mortgage, the stock could be listed as an asset and if it is
substantial, the terms of your loan couled be greatly improved.  You
will likely need some sort of  accountant's statement or something
that proves the value of your shares.  You can get great credit if you
can show that you don't really need it - this seems to be the general
rule.  Depending upon your individual situation, listing the shares as
an asset might get you a credit card with a large limit and reasonable
rate.  This is a way of using those shares as an indirect kind of
collateral.  In today's environment, lenders are in fierce
competition, and they want to lend you money very badly.

But borrowing directly against the shares puts the bank at greater
risk.  It is difficult for them to know that the value of the shares
will not drop substantially - turning the collateralized loan into an
uncollateralized one.

In theory, yes. In practice, as daytrader76 suggests, in your case
probably not. So long as you can provide collateral and satisfy the
lender of the collateral's value, the lender may lend you a portion of
its value. The key element here satisfying the lender of the
collateral's value and, as part of that concept, is satisfying the
lender that if the loan goes bad, that it can recover its investment
by liquidating the collateral. That's much more difficult to do with
privately-held or seldom-traded stock. Having said that, IF you have a
good pre-existing relationship with a bank, you might be able to
persuade it to accept the stock as collateral. In that case, it would
be making the lending decision based both on the value of the stock
(assuming you could clearly establish it) and on your good credit
history and your value to the bank through your other relationships.

This is a case by case matter.  Most low price stocks are very
volitile and wouldn't make a very good colateral source, but in order
to find out talk with a person banker at your local bank.  Hometown
bank in fond du lac, WI excepts stock as collateral, bu case by case. 
You may also want to consider buying options to reduce your risk in
the stock, or profit from future up side of the stock today.