Irishman McQuaid was relying on Swiss Cycling's support as he bids for a third term as UCI chief but the withdrawal of their nomination leaves him needing a change to the international federation's statutes to enable him to stand for the presidency.
The back-pedalling over McQuaid has left divisions within Swiss Cycling that provoked Chassot's decision.
"Richard Chassot handed in his resignation from the presidency of Swiss Cycling with immediate effect," said the federation in a statement.
It said the decision was related "to the differences within the board of directors about the nomination of Pat McQuaid."
McQuaid was counting on the Swiss nomination, having already seen his homeland withdraw its backing for his candidacy.
According to UCI statutes a candidate must be supported by his home federation.
After the Irish withdrew their backing, McQuaid turned to the Swiss federation, which qualified him for candidacy on the grounds of his residency in Switzerland, where the UCI's seat is.
But now, to stand against British Cycling president Richard Coockson at the September 27th elections in Florence, McQuaid will need the UCI to change its rules.
He has the support of the Thai and Moroccan federations while the Malaysian federation has proposed an amendment to the current statutes that would allow a candidate to stand as long as he had the backing of any two federations.