Suppose all the data is positive. Squaring it means that the bigger values in the hump will get multiplied by a bigger number than the smaller values in the tail. Doesn't that just exacerbate the problem?
Numeric example: 10% of the values at 1, 90% of the values at 100. Squaring it makes the skew even bigger.
10% at 1 and 90% at 100 isn't the greatest example of left skew - it's binary. Let's look at something a bit more typical of this. For instance, the SAT scores at a very prestigious university would likely be left skew, since there is a ceiling of 800 (on each section). I found this site which claims the mean is 750, 25%tile is 700, 75%tile is 800. So, let's fake some data and find skewness. R code (stuff after a # is comment, in this case, the result:
library(moments)
set.seed(1234)
SAT <- rnorm(1000, mean = 750, sd = 80)
SAT[SAT > 800] <- 800
quantile(SAT, c(.25, .5, .75)) #696, 749, 799
skewness(SAT) #-0.96
SATsq <- SAT^2
skewness(SATsq) #-0.74
So, squaring did lessen skew. Of course, that variable wasn't that skew. Let's suppose that there is some university that is usually as selective as Harvard, but waives all requirements for their sports teams (Harvard does NOT do this). The president of Harsports U. is a sports fanatic.
sports <- rnorm(200, mean = 400, sd = 50)
harsports <- c(SAT, sports)
skewness(harsports) #-1.3
harsportssq <- harsports^2
skewness(harsportssq) #-1.03
Still not that much negative skew. But squaring did reduce it a bit.
EDIT: We can see this, perhaps, with density plots, although I admit they aren't the clearest indication:
plot(density(harsports))
lines(density(harsportssq/800), col = 'red')
which yields
But to really see it, you can look at the formula for skew and see what squaring would do.
