I did some cross searches of the names surrounding the styles and people you mention...have to admit, first impression isn't good - in one way or another, most leads are dead-ends and many seems to tie in with various soke-ship councils.

If it's a 'Ryu', then that means it must be or have been studied in Japan/Okinawa at one time or at present. I'm guessing it's not, nor ever has. If the founder and transmitter of it can't be located, then there is no sense of even trying to claim legitimacy.

If there is any truth to "Kensenzue Yamaeugue" even being a real person. The only indication of what might have been passed down is the kata of the system. It appears Pinan & Naihanchi, Kusanku and Seiuchin and Sanchin.

However, reading here:
sortof gives the sense of some history behind the style, at least a generation anyway.

sifting thru, I got the sense of the style being from a Shotokan-based art of Korean origin....then later other Budo Arts added.

The name Oscar Adams comes up here:
which references him tied to the Armed Forces Judo Association as yudansha in the Navy during the 60's.

I'm envisioning this complete guess: Ray Flowers, Terry James, and Oscar Adams all served overseas and brought back Korean Shotokan, Judo and perhaps an Okinawan art such as Goju. They perhaps collaborated/shared and went their own ways. Oscar Adams teaches Williams during the 80's, Mr. Williams later corroberates with others and forms/names Eugue Ryu in the 90's.

most likely, it IS "made up". In fact, Mr. Williams' own daughter says so in the link I gave.

Let me just start by saying that floor work is a very important part of Eugue Ryu karate. It teaches discipline, focus, and endurance. If you do not see it that way, then you are not learing a true Eugue Ryu style and should look else where. My father, Steven Williams, helped start this style in KC and taught Mr. Jones, Mr. Hess and Mr. Baswell. By what I have heard over the last few years, some of those people are holding true to what they were taught, others were not. ...

Thats not a bad thing. It's only a 'bad' thing if people lie and join sokeships for rank to try and legitimize on paper what they do and have learned (or not learned) on the mat. "Making up" an Art by making it your own is never the problem - I'm sure Eugue Ryu is as in-depth as any other has to offer. and I'm sure, like all other Arts...there are good and bad eggs.