The claim that 62% of Americans believe Republicans will 'win' the government shutdown is true because a Harvard poll reported by The Hill found that 62% of respondents thought Republicans would come out ahead in the shutdown standoff.
62% of Americans Believe Republicans Will ‘Win Government Shutdown,’

Trump and Republicans are being blamed more than Dems for government shutdown, early polling finds - The Independent
Source: independent.co.uk
Key Evidence
A Harvard poll reported by The Hill found 62% of Americans believe Republicans will win the shutdown.
thehill.comthehill.comShutdown poll: Americans blame Republicans, want deal
Other polls show Republicans are blamed more than Democrats for the shutdown.
independent.co.ukindependent.co.ukTrump and Republicans are being blamed more than Dems for government shutdown,...
thehill.comthehill.comMore Americans blame Trump, Republicans for shutdown than Democrats: Survey+1 more
This indicates that despite blame, many expect Republicans to come out ahead politically.
What the Evidence Shows
Multiple recent polls from reputable sources indicate that while a majority of Americans tend to blame Republicans, particularly Trump and GOP leaders, for the government shutdown, there is also a significant perception that Republicans will ultimately 'win' or come out ahead in the shutdown conflict. Specifically, a Harvard poll cited by The Hill explicitly states that 62% of Americans surveyed believed Republicans would win the shutdown, compared to 38% who thought Democrats would prevail.thehill.comthehill.comShutdown poll: Americans blame Republicans, want deal
Other polls consistently show Republicans receiving more blame for the shutdown than Democrats, but this does not contradict the perception of their eventual political advantage. The data reflect a nuanced public opinion where blame and perceived victory are distinct concepts. The claim is supported by direct polling data from a credible academic source and reported by multiple mainstream news outlets.independent.co.ukindependent.co.ukTrump and Republicans are being blamed more than Dems for government shutdown,...
thehill.comthehill.comMore Americans blame Trump, Republicans for shutdown than Democrats: Survey+2 more