According to Cane Fires: The Anti-Japanese Movement in Hawaii, 1865-1945:
The Speak American Campaign was launched on October 12, 1942, with much fanfare in the white press. The Service Committee distributed and displayed posters and stickers admonishing the Japanese to "speak American," and committee members held community meetings to organize English-language classes for adults. In a radio address, Shigeo Yoshida explained the need to "speak American".
The war-time article Speak American by Shigeo Yoshida, principal Ala Maana School, can be found at that Hawaii Educational Review link:
THE "Speak American" campaign sponsored by the Emergency Service Committee has the endorsement and cooperation of military and civilian leaders. It is worthy of the support of all who need the admonition to "Speak American" or who are in a position to help such people learn and use the language of America. The idea of speaking English is not new. The schools and other educational agencies have been working on it for years. It is particularly important in Hawaii where we have many large groups of foreign-language-speaking people...
If this were Japan, which it is not and we know it will never be, the use of the English language would certainly not be tolerated as the use of the Japanese language has been here. That is one of the differences between the American and the Japanese ways of life. But we should not take advantage of the tolerance which is part of America by using the language of the enemy in situations where its use is not desirable or absolutely necessary...
I wish to mention one other reason for urging the use of English. We think in the language we speak. I believe it is more important than ever that we in Hawaii not only speak but think in the language of America. Our slogan might be: "Speak, think and act American." ...
And Shigeo Yoshida was correct, this was not new.
In the June 1920 Popular Educator, Vol. 37, there are photographs of posters:
Another poster has 3 children holding cards saying "Be American", "Think American", "Speak American".
The journal Primary Education November 1919, page 604 has a list Slogans for Better American Speech Week with patriotic and more general suggested slogans including:
Speak the language of your flag.
American Speech means American Loyalty
The American Flag, The American Home, The American Language
American Speech for American People
Be patriotic in thought, in deed, in speech
American Speech means American unity
Daughters of the American Revolution Magazine, Volume 54 says that between February 1919 and February 1920:
We also had posters printed containing the words, " Speak American." These are to be hung in public places.
Below is a poster as published in the 15 December 1918 Chicago Daily News
See also the 19 January 1941 Chicago Tribune NW edition, part 3, page 1 article Parks Sponsor New Lessons in American Way
new campaign to "Act American; Speak American; Be American"
As far as the first poster in the OP, an Italian History Society link indicates in was provided by the Center of Military History.
A thesis gives a different source:
HWRD Scrapbooks. (Blackout, Rumors and Propaganda, Camouflage 1942 February- 1945 July 23, Folder 1)
where "HWRD" means Hawaii War Records Depository