What does “Give a chicken in every pot” mean?

This is attributed to King Henry IV of France, who reigned from 1589 to 1610, and was reported to have said he wished for the peasantry

Un poule au pot le dimanche, A chicken in every pot on Sunday

Henry picked chicken, and it's been chicken ever since.

This is reported to be a slogan of Herbert Hoover's 1928 Presidential campaign:

A chicken in every pot and a car in every garage.

But apparently the candidate never said it. It appeared in newspaper ads bought by Republican supporters.

Huey Long, the Louisiana populist (or demagogue, depending on your point of view) is supposed to have adopted the slogan for his 1932 campaign for governor:

A chicken in every pot, and every man a king

Given the fact that Hoover presided over the start of the Great Depression, the slogan is often used ironically to comment on politician's overblown promises of prosperity that never materialize. In your example, one of the Republican candidates is criticizing his fellow candidates for their economic plans, which consist mainly of cutting government spending and cutting taxes. Since they never say what programs they'll cut, it's unlikely they have any real plans to do so. They do have details about their tax cuts. The result would be to reduce revenue without reducing expenditures, a recipe for a disastrous increase in debt. Kasich is saying these plans are fantasies that would never be implemented, just like the promises of chicken for everybody.

"I don't do ," (where X is a noun) is a common, slangy locution for "I don't do the activity concerning X":

I don't do drugs (I don't take drugs.)
I don't do mornings (I'm not effective at anything before noon.)
I don't do relationships. (I don't get emotionally involved.)

And so on.

The exact quote is "If God keeps me, I will make sure that there is no sharecropper in my kingdom who does not have the means to have a chicken in the pot every Sunday!" King Henry IV brought a period of peace and prosperity to France, partly because he instituted policies that benefited farmers and the common people.

The use of the word chicken simply gravitates more towards the time period and circumstance for which the phrase was coined.

To answer the second portion of your question: The phrases "Don't do substance" and "Don't have substance" aren't really different at all. The reporter/writer is simply referencing the fact that it appears all too common, among G.O.P candidates, to not have substance in their debates. In short: One is acting out - voicing out -"something" with substance and the later is more of a description.

Source: American Thinker