Is EOF always negative?

Yes, EOF is always negative.

The Standard says:

7.19 Input/output
7.19.1 Introduction

3 The macros are [...] EOF which expands to an integer constant expression, with type int and a negative value, that is returned by several functions to indicate end-of-file, that is, no more input from a stream;

Note that there's no problem with "plain" char being signed. The <stdio.h> functions which deal with chars, specifically cast the characters to unsigned char and then to int, so that all valid characters have a positive value. For example:

int fgetc(FILE *stream)

7.19.7.1
... the fgetc function obtains that character as an unsigned char converted to an int ...

EOF is always == EOF. Don't assume anything else.

On a second reading of the standard (and as per some other comments here) it seems EOF is always negative - and for the use specified in this question (line number or EOF) it would work. What I meant to warn against (and still do) is assuming characters are positive and EOF is negative.

Remember that it's possible for a standard conforming C implementation to have negative character values - this is even mentioned in 'The C programming language' (K&R). Printing characters are always positive, but on some architectures (probably all ancient), control characters are negative. The C standard does not specify whether the char type is signed or unsigned, and the only character constant guaranteed to have the same value across platforms, is '\0'.