Do you hold the stock in a retirement account (IRA, 401(k), etc.)? If you do, there will be taxes and penalties to take into account if you cash out. If you must access funds from a restricted account, it may be better to take a loan from your 401(k), avoiding the taxes and penalties, but obligating yourself to replenish the 401(k) (which you might intend to do anyway once your husband gets a job). (If you just want to diversify, you can sell some shares and buy different stocks/bonds/money funds within the account without penalty.)
If you do not hold the stock in a retirement account subject to any special tax benefits, then you will not face any penalties if you sell, but you will realize taxes on the gain (probably at the capital gains tax rate).
In any event, I do not think that “paying down the house” sounds like a good idea. What you want, as others have noted, is liquidity–extra cash now for living expenses. Paying down principal is exactly the opposite–tying your money up in the house. Even if you pay the house off completely, you trade your current liquidity (the stock) for liquidity in small amounts over time (the relief from monthly mortgage payments). Why not, if the stock is not held in a restricted account, instead just sell off enough stock each quarter (or other timeframe that matches your need for cash) to pay those extra living expenses? Or, if you do need to diversify, sell all of the stock now, keep a portion in a short-term money market-type account, and reinvest the remainder in a no-load broad-based mutual fund (or perhaps better, a bond fund since it sounds like you do not have a lot of time before investment and this is less risky than a stock fund).
One more thing: have you refinanced your mortgage lately? While banks don’t like to give terms of less than 10 years on fixed rate loans, and it doesn’t sound like you’d like a term any longer than you have now, rates are significantly lower than they were only 2-3 years ago, and you should not be paying more interest than is necessary. Your husband’s lack of a job may or may not matter to the bank, depending on the equity you have in the house and the amount you earn at your job.
I am sorry to hear that you are having a rough time. Hope a job comes soon.