Squared return is basically a measure (the simplest measure) of volatility, it shows how big the day's return is without looking at whether it is positive or negative. Periods of time with big squared returns are volatile periods (The Great Recession of 2008-2009 for example).
So the regression you propose basically shows how volatility of a stock today is affected by today's index volatility as well as yesterday's volatility in the index and in the stock. The lag coefficients will presumably be positive, showing that volatility movements are persistent from one day to the next. Big vol yesterday predicts big vol today and vice versa low vol yesterday is usually followed by low vol today (with exceptions of course, surprises do happen).
These kinds of regressions are often performed in volatility studies. They are also related to volatility models like GARCH and so on that try to track volatility over time. This kind of research has shown that market volatility is not constant but changes in predictable ways.