Not familiar with that genus (we don't get them this far north), but I was curious so I looked over some references. I think you can rule out Phemeranthus calcaricus because it occurs on limestone outcroppings. Can also say that from experience whenever I think I found a new species or hybrid it was either an oddball specimen or I was misinterpreting the morphological characteristics in a key. While the sepals should be deciduous, are you looking at mature capsules or just spent flowers? Cannot find reference anywhere to the shape of the style

You can look over some specimen images of P. mengesii here:

http://www.floraofalabama.org/SpecimenDetails.aspx?PlantID=2978

You may also find more information using the original name, Talinum mengesii. The type specimen was originally in the Herbarium of St. Bernard College but was apparently donated to Auburn University in 1995.

Thanks.
Here is how the sepals are on this plant: When it flowers the sepals are green, when the pink flower petals wilt the sepals are still green, when the fruit starts to swell up between the two sepals they start to turn a little brown, when the fruit is mature and at full size the sepals are either very shriveled and brown or they blow off.
I'm not sure, but I was thinking the sepals are deciduous if they fall off right after flowering, But I might be wrong.

I am leaning more toward it being: P. mengesii

But probably the biggest mystery to me is the stigma. This plants stigma is clearly distinctly 3-lobed.

Flora of North America stigma description:
Phemeranthus calcaricus: stigma 1, distinctly 3-lobed.
Phemeranthus mengesii: stigma 1, subcapitate (almost head or caplike).

I know with Phemeranthus parviflorus (definitely not my plant) Flora of North America gives this: stigma 1, subcapitate (almost head or caplike), sometimes stigmas 3-lobed, or triangular). Could Flora of North America missed the fact that Phemeranthus mengesii could PERHAPS have a 3 lobed stigma on some rare populations?

I drew a picture of what the stigma looks like (the very clear distinctly 3-lobed stigma):
{{gwi:269406}}

If you have access to journals it would be worth looking for some of the articles written by Wolf. For example

The Status of Talinum in Alabama
W. Wolf
American Midland Naturalist
Vol. 22, No. 2 (Sep., 1939), pp. 315-332

With regard to the stigma Wolf says:

"The Stigma: This structure deserves special discussion in connection with all three Appalachian species. It also serves as an indicator of degree of affinity. Among the three species T. mengesii is the one that exhibits a fair degree of constancy in the form of stigma, which is rather consistent with the character of a supposedly extracted species. Here, the stigma is either actually capitate, or at least so in appearance; in the latter instance the short lobes are erect in anthesis, but usually slightly gaping."

This suggests that the stigma can have lobes provided they are erect. The question is can they be as long as you show in your illustration?

Interestingly, Wolf discusses at length the possible relationship between T. teretifolium and T. mengesii (he thought T. mengesii decended from T. teretifolium). The description for Phemeranthus teretifolius in FNA states:

"Phemeranthus teretifolius is an allopolyploid derived from P. parviflorus and P. mengesii, both of which it evidently outcompetes within its southern Appalachian range (M. E. B. Carter and W. H. Murdy 1985; W. H. Murdy and M. E. B. Carter 1985, 2001)."

Thanks, that very interesting, so I guess that might explain it.
Do you know what he means by they are erect in anthesis? (I know that means in flowering, but does he mean the lobes are erect just during part of anthesis or are they always erect through the entire flowering or anthesis)? I guess he means through the entire flowering. I still have the Stigma I cut off to observe from 5 hours ago, and the lobes are still erect.

From a few feet away it's hard to see the lobes, but if you get up close (a foot or less) it's easy to see the 3 perfect lobes (at least for me).

I may have drew the lobes a tad too long. If you cut off the top 1/3 of the lobes in my drawing it might be more true to how it really is.

And do you know what he means by "but usually slightly gaping"?

I'm guessing that's the space between the lobes ("stigma sinuses"). If that's what he means, they seem more deep than slightly to me (But I don't know his definition of "slightly").