I cannot disprove it. But as a JFK backer and follower I can tell you it is not typical JFK language and is a grammatical misquote if factually true.
The reference to "high office of the President" does not refer to the VP or the administration, only to the President and his staff.
It is exceedingly unlikely that JFK would have made such an accusation about a Democratic president (and the last Republican president left office 30 years earlier) and not have attracted widespread publicity at the time.
As a newspaper reporter at the time I can tell you there was no such publicity.
Further JFK would not have used the term "the American's freedom." As stated it is singular possessive and would refer to only one American. If there was no apostrophe used JFK was too much of a stylist to use the article "the" in that term.
And, finally, the term "the citizens of his plight" is also grammatically incorrect. The plural "citizens" is not used with the singular "his."
So in answer I would say the quotation is such mangled English that is is virtually certain JFK did not say it.