ANYMAN: Got it right with the large width highlighter.
DOGROCK: If you draw a line around the earth it may be considered a spiral
PS: sorry about dropping a question and taking off (no offence intended).

turning the paper over may work, possibly easier if the circles were arranged as a Rhombus (as I ‘think’ this would give a straight line at an angle, without the need to adjust the width of the line, I'm probably wrong), however, it could be argued that this would form a very narrow spiral around the thickness of the paper.
The same argument of a spiral (although more valid) could be made for forming the paper into a cylinder and drawing the line around the circumference.

However. There is a possible alternative, although I will leave it to you guys to prove it right or wrong, hopefully with some form of mathematical proof.

Problem: to connect 9 circles arranged in a square, with 1 straight line, without folding or tearing the paper, and without the drawing implement leaving the surface of the paper.

Possible solution:
1) take a piece of A4 paper, cut a lengthwise strip about 1 inch in width.
2) draw the 9 circles on it in the shape of a Rhombus (or ordinary square).
3) Twist the paper through 180 degrees
4) take 2 small pieces of sticky tape,
5) join the 2 ends of the paper together (using 1 piece of tape for 1 edge and 1 piece of tape for the other edge, leaving a gap where the paper is not covered by tape).
6) you now should have a Mobius strip/band.
7) take your pen and connect the first 3 circles (lengthwise along the strip, going with the angle of the Rhombus).
8) continue around the strip/band until you have connected all of the circles.

Points of discussion, (bearing in mind that the strip/band has only 1 edge and 1 side).

1) is there a fold in the paper? To me this is not a fold, but a curve. (does anybody out there disagree with this distinction)

2) is the line a spiral? (in a mathematical sense)

3) is the line straight ( in a mathematical sense)

4) is it something else

5) would the use of the Rhombus make any difference whatsoever (to the possible validity of this possible solution).

Somebody out there must know the answers to the above questions, (they are probably quite obvious answers to you guys), so please prove me right, or wrong, as the case may be.