BJT as a switch: what am I doing wrong?
I don't think the input is a square wave that has a negative half cycle. I think the dotted line between \$E_C\$ and \$-E_C\$ indicates a positive threshold voltage, and if the input falls below that threshold then the output will go high because the base current is too low to keep the transistor in saturation. There is no way that \$V_{CE}\$ can be positive, as indicated, if the only circuit supply voltage is negative.

Since this looks like homework I'll leave you to do the analysis.

It's not a very good schematic, in my opinion. Let's hope your textbook provides a good narrative to go along with it.