It is the same construct - simply test for empty strings rather than NA:
Try this:
df <- df[-which(df$start_pc == ""), ]
In fact, looking at your code, you don't need the which, but use the negation instead, so you can simplify it to:
df <- df[!(df$start_pc == ""), ]
df <- df[!is.na(df$start_pc), ]
And, of course, you can combine these two statements as follows:
df <- df[!(df$start_pc == "" | is.na(df$start_pc)), ]
And simplify it even further with with:
df <- with(df, df[!(start_pc == "" | is.na(start_pc)), ])
You can also test for non-zero string length using nzchar.
df <- with(df, df[!(nzchar(start_pc) | is.na(start_pc)), ])
Disclaimer: I didn't test any of this code. Please let me know if there are syntax errors anywhere