This answer assumes that the person we are looking for is identical with the author of the following paper:
John Thomas, "A regular space, not completely regular." The American Mathematical Monthly, Vol. 76 (1969), No. 2, pp. 181-182.
The paper gives the author's affiliation as New Mexico State University. I found two mentions of Dr. Thomas in AMS publications from around the time of publication:
Notices of the American Mathematical Society, Vol. 16, No. 2, February 1969, p. 394:
Dr. John D. Thomas, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88001.
Notices of the American Mathematical Society, Vol. 17, No. 6, October 1970, p. 872:
Dr. JOHN D. THOMAS of New Mexico State University will be on sabbatical leave during the fall semester of 1970. He will spend his leave at Los Alamos Scientific Laboratory, New Mexico.
We also find Dr. Thomas mentioned in various annual reports of the mathematical sciences department of NMSU.
From the 1965-1966 report we can infer that Dr. Thomas joined the faculty of NMSU around 1960:
While the new staff added for next year are of very high quality and will further strengthen the group, it should be pointed out that of the twenty-four staff members presently hired for next fall, only Drs. Giever, E. Walker, Thomas and Ader will have been on the staff more than four years
In the 1966-1967 report he is is listed as a conference attendee:
Dr. John D. Thomas, Second Annual Conference on Pure and Applied Mathematics, (State).
In the 1968-1969 report he is listed (p. 221) as:
John D. Thomas, Project Directors of NSF Summer Institutes Meeting (national). Holiday Mathematics Symposium (national).
In the 1970-1971 report he is listed (p.184) as:
Thomas, John D., Pi Mu Epsilon Faculty Sponsor, Reviewer for Zentralblatt fur Mathematik
In the 1972-1973 report he is listed (p. 182) as:
John D. Thomas, Reviewer for Zentralblatt fur Mathematik. Chairman, Departmental Committee for Applied Mathematics.
In the 1977-1978 report he is listed (p. 206) as:
J. D. Thomas, Reviewer, Zentralblatt für Mathematik. Faculty advisor, Alpha Tau Omega Fraternity. Director, intercollegiate regional bridge tournament, NMSU Bridge Club. Chairman, Computer Advisory Group.
In the 1983-1984 report, he is listed (p. 264) as:
J. D. Thomas, Chairman, Department of Mathematical Sciences Undergraduate Majors Committee (spring), [...] Chairman, doctoral committee, mathematics (one). [...]
From the Mathematics Genealogy Project we learn that Dr. Thomas advised nine Ph.D. students at New Mexico State University between 1965 and 1988, and that his full name is John David Thomas. We further see that he received his Ph.D. from the University of Oklahoma in 1959. The title of his dissertation was "A Local Coefficient Cohomology Theory for Lattices". This is confirmed by a catalog entry at the library of the University of New South Wales in Sydney, Australia, which holds a copy of this thesis on microfilm.
There are two mentions of Dr. Thomas in minutes of the meetings of the Board of Regents of the University of Oklahoma.
June 9, 1955:
GRADUATE ASSISTANTS [...] John David Thomas, Department of Mathematics, \$1,500 for 9 months, September 1, 1955
January 17, 1957:
John David Thomas, title changed from Graduate Assistant to Instructor in
Mathematics, salary changed from rate of \$1,500 for 9 months, 1/2 time,
to rate of \$3,624 for 9 months, full time, January 16 to June 1, 1957.
So far I have identified the following publications by Dr. Thomas:
John D. Thomas, H. Ralph Lewis, and K. J. Melendez, "Computation of Fourier integrals of exponentials of truncated Fourier series." Journal of Computational Physics, Vol. 12, No. 4, August 1973, pp. 462-470
John D. Thomas and Joseph D. Zund, "A note on the theory of immanants." Rendiconti del Circolo Matematico di Palermo 28, 143-150 (1979)