JOEL - I have never heard of any official word from the Church leaders that suggested that for personal reasons we might be able to observe the Sabbath on any other day except Sunday.
The key revelation giving the pattern, scope, and purpose of Sabbath observance came to Joseph Smith on August 7, 1831, a Sunday:

"And that thou mayest more fully keep thyself unspotted from the world, thou shalt go to the house of prayer and offer up thy sacraments upon my holy day;" (D&C 59:9)

There is one exception that I know of as explained in The Encyclopedia of Mormonism:
"Since 1950 the Church has adjusted to issues of a cultural and political nature. One example is the First Presidency's decision to allow members to hold Sabbath services, customarily reserved for Sunday, on the day of worship designated by local religious tradition: Friday in Muslim countries and Saturday in Israel."

Besides this exception I don't believe it is proper for anyone to designate another day of the week as their Sabbath day, simply for conveinience of their work. If the person does not have to work the entire day, they could attend another ward's sacrament meeting that is at a better time. They could also get permission from the Bishop to hold their own Sacrament in their own home on Sunday when they are not at work.

Here is some information about this from the Church Handbook:
"Every member needs the spiritual blessings that come from partaking of the sacrament. Occasionally members may be unable to attend sacrament meeting because they are confined to a home, nursing home, or hospital. The bishop may assign priesthood holders to prepare, bless, and pass the sacrament to these members. Occasionally members may be unable to attend sacrament meeting because of distance to the meetinghouse. Under unusual circumstances, the bishop may give authorization for a sacrament service to be held away from the meetinghouse. The priesthood holder whom the bishop authorizes to conduct the service must be a priest in the Aaronic Priesthood or hold the Melchizedek Priesthood. He also must be worthy to bless and pass the sacrament. The bishop's authorization is also required if the service will be attended by members of more than one family. The priesthood holder who directs the service reports to the bishop when the service has been held."
See also the following Church web site: http://www.lds.org/library/display/0,4945,13-1-1-6,00.html

Of course the bottom line is that we should avoid working on the Sabbath if at all possible. Here is a quote from Spencer W. Kimball:
"It is true that some people must work on the Sabbath. And, in fact, some of the work that is truly necessary—caring for the sick, for example—may actually serve to hallow the Sabbath. However, in such activities our motives are a most important consideration.
When men and women are willing to work on the Sabbath to increase their wealth, they are breaking the commandments; for money taken in on the Sabbath, if the work is unnecessary, is unclean money…. "

I am sure a quote like this might not sit well with those who find they must work on Sunday. But I believe proper observance of the Sabbath has a lot to do with the attitude one has as they spend the day, whether they are working or not. We should still maintain the spirit of Sabbath worship in our hearts as much as possible.
Besides our profession we sometimes find ourselves needing to do other things we should not do on the Sabbath. President Harold B. Lee was fond of the saying that, "It's all right to pull the cow out of the mud on Sunday, provided that you don't push him in on Saturday night."

As far as a person's worthiness for baptism is concerned, I don't know of anyone who has been denied baptism simply because their profession forced them to work on Sunday. During the conversion process, while they are taking the discussions, an investigator is usually taught about the Sabbath and is encouraged to try and get Sundays off if possible. But if they have repented of their sins and understand the concepts of the Gospel and the responsibilities and promises they make at baptism, and are worthy in every other way, they should be allowed to be baptized. I know of no regulations that say otherwise.
Everyone must establish their own personal relationship with God in order to help them know what to do for their own unique situation so that they feel right about what they do on the Sabbath day.