Anne, the tests associated with Elizabeth St. John as shown on that St. John Genealogy Project website are for autosomal DNA only. Nothing is impossible, but the likelihood of informational accuracy using autosomal DNA for someone born in 1631 is pretty close to it. Exhuming and testing the remains would be the only way it could be done, and even that might be inconclusive.
Biologically and mathematically, the reach of autosomal DNA for accuracy in genealogy is about 5 or 6 generations; 7 might be possible, but in my 21 years of doing this stuff I've never seen an instance of it that I couldn't dispute.
If we use a lenient 28 years as an average generational interval, Elizabeth would be approximately 12 generations back from living test-takers. That would mean the test-takers are separated by 48 unique meiosis events in which the parent's DNA recombined what they received from their parents, and a child would then get 50% of the mother's DNA and 50% of the father's.
It theoretically takes a couple having more than 7 children for them to pass down 99% of their DNA to future generations. That many children then need to produce more than 7 of their own at each generation or we lose chunks of ancestral DNA. U.C. Davis population geneticist Graham Coop has shown that, from 12 generations back, there is about a 90% chance that a descendant would have no measurable blocks of autosomal DNA from any given ancestor.
That mechanism for assuring adequate genetic diversity within a species means that the theoretical amount of Elizabeth's atDNA carried by a test-taker, 12 generations later, would be approximately 0.006%. There could be zero; there could be a little more. Our popular DNA tests can't accurately identify sharing amounts much lower than about 10cM. That 0.006% of sharing would be equivalent to roughly 0.42cM.
The Williams Lab at Cornell has run simulations of cousin-matching probabilities out to 8th cousins. In Elizabeth's case we would be looking at 10th cousins. A study by Brenna Henn, et al., in 2012, attempted to take the estimates out to that 10th cousin level. They offered 0.002% as the chance that any two 10th cousins might share a detectable amount of matching DNA regardless from which ancestor it stemmed.
To get a very broad estimate of two 10th cousins sharing the same measurable bit of DNA from the same ancestor, we can convert the percentage to a decimal probability (0.00002) and square it, for a result of 4x10-10, or 0.0000000004. For a factor of magnitude allowance as a margin of error, let's multiply that by 10, for 4x10-9 = a 0.0000004% chance, or 1 in 250 million. The odds of winning a Mega Millions Powerball jackpot aren't much better at about 1 in 302 million. 
The data at the St. John Genealogy Project website cites segments as reported by GEDmatch of 3.3cM, 2.3cM, 2.7cM, and 5.3cM. Some personal research using results from whole genome sequencing has convinced me that at GEDmatch--and using default matching parameters--segments smaller than 20cM may be false up to 60% to 68% of the time. In my opinion, citing segments as small as those shown in the St. John project can never be accurately validated without comparing actual WGS test results and, even then, segments that small simply wouldn't be traceable to a specific ancestor.
We all carry many, many DNA "echoes" that come from founder and localized populations. These are pieces of DNA that resulted from generations of a specific population living nearby each other. In fact, without that the DNA testing companies couldn't even attempt to provide us with "ethnicity" results. We most definitely inherited that DNA by way of an ancestor, but that origin was so long ago that many people carry the same bit of DNA from some ancestor born before the genealogical timeframe and before there were records we could use to try to identify the earliest progenitor.
For example, AncestryDNA pegs me currently at 30% Scottish and 17% Irish. Y chromosome data compiled from over 30 test-takers adds support to the notion that my patrilineal line, at least, was centered around the southwestern corner of Scotland as far back as circa 900 CE. I carry tiny bits of that population-level autosomal DNA, as do many thousands of other people living today. But it will never be possible to identify the earliest ancestors from who it came.
My opinion is that any autosomal DNA information that purports to trace back to Elizabeth and the early 17th century should be discarded as non-evidentiary.
