If my husband was deeded a cabin in his name only, would it automatically become my property if he were to die?

If it is in his name only, then the property would become part of his probate estate upon his death. Then depending on the terms of your husband's Will (if he had one) you would in all likelihood be named as his beneficiary; if you were not named in his Will, then by virtue of something known as the "elective share", you could still lay claim to a part of the estate.

If your husband did not have a Will, then his estate would be distributed according to the "intestacy"  laws of the state in which he was domiciled as of his death. This is typically 1/3 - 1/2 of the assets to the surviving spouse with the remainder to be split among the deceased's children, if any. Otherwise the entire estate goes to the spouse.

In order to take over the property automatically, you could have him put your name on the deed as his wife "with right of survivorship"; this form of ownership is known as a "tenancy by the entirety". This would insure ownership rights (100%) without the time and expense of putting the property through probate.

There are other ways for you to takeover the property upon his death. At this point you should speak directly with an estate planning attorney as to your options.