It's true that Chick-fil-A (and a number of food manufacturers) use the ingredient dimethylpolysiloxane in food, sometimes as an anti-foaming agent.
It's false to say that dimethylpolysiloxane is Silly Putty.
The tweet was not the first time the claim about Chick-fil-A's use of dimethylpolysiloxane surfaced. In October 2013, an earlier version titled "You Won't Believe Where Silly Putty Is Hiding in Your Food" targeted several large food companies for their use of dimethylpolysiloxane.
While the focus narrowed to Chick-fil-A in later circulations, the pattern of the claim was similar to earlier food ingredient warnings.
First, loudly warning that "you won't believe" what various companies are "hiding" from you in readily available ingredient lists. Then, petition for public pressure to "shame" companies into using only pronounceable ingredients.
The conflation of dimethylpolysiloxane with the elastic-like Silly Putty toy amplified the rumor. This is possibly because anyone who's attempted to remove the latter substance from a carpet knows it's virtually indestructible (particularly when the carpet is brightly colored). Folks have extrapolated that the notoriously difficult-to-extricate goop is not a health food and may not be fit for consumption at all.
It is true that Silly Putty contains dimethylpolysiloxane at a concentration of roughly four percent. However, other polymers and silica are the toy's primary ingredients. Ninety-six percent of Silly Putty is composed of ingredients other than dimethylpolysiloxane, so the comparison is certainly overstated and inaccurate.
Also notable is the frequency with which dimethylpolysiloxane appears in food products overall. Deemed safe in concentrations up to 250 parts per million by the Food and Drug Administration (FDA) and approved for international use by the World Health Organization (WHO), dimethylpolysiloxane is included on ingredient lists from a number of food manufacturers. McDonald's, Taco Bell, Whataburger, Coke, and KFC all use dimethylpolysiloxane in their menu items. Some versions of the rumor claim that the ingredient is banned in the United Kingdom for safety reasons, but that is not true.
Ultimately, it is true that Silly Putty and Chick-fil-A products share a common ingredient, albeit both in small quantities. But the primary objection to dimethylpolysiloxane appears to hinge on the fact that the word itself is somewhat intimidating and sounded "chemical-y." Not only is it present in a number of foods widely available for sale, but no specific objections documenting a lack of safety have been raised in any of the circulated claims. In lieu of legitimate concerns about the safety of dimethylpolysiloxane, the rumors simply have targeted its use in a common household toy, which in and of itself does not constitute cause for alarm.