Using a 401(k) distribution to pay down a home mortgage is usually not a good idea. In addition to regular income tax, if you are under age 59 1/2 you get hit with a 10% tax penalty. You only want to consider this option if you are at risk of losing the house under foreclosure.
When you receive the distribution, there would be some tax withheld. It is seldom the correct amount. The distribution is reported on your tax return and the balance of the tax is paid at that time.
If your 401(k) is getting distributed to you because of the change in jobs, consider rolling it into an IRA. There would be no tax consequences to that move.

At age 68 can you invest your 401K monies directly into buying a house and not pay taxes on it?

There is no way to avoid income tax on distributions from a 401(k).

At age 68, there would be no early distribution penalty, and you can use the money to buy a house if you wish. But the distribution is reported on your tax return, and you do pay regular income tax on it.

You might want to consider taking part of the amount you need in December, and the rest in January. Depending on your tax situation, that might allow you to limit the tax pain. You'd have to talk to someone who could review your entire tax situation to see if that was worth doing.