Why can scalars have a sign?
It has to do with what point you define as a reference, what do you call (define) as zero? When talking about temperature it depends which unit you measure in. In example, when using Celsius, zero is defined as the freezing / melting point of water (under normal pressure etc). Do we know anything colder than that? Yes we do. The only way to expres these values < 0 is by using negative numbers. When using Kelvin however zero is defined as the absolute zero temperature and therefor does not know any negative values.
After all - the plus and minus indicate the direction of the scalar on a one-dimensional axis.
The numbers (signs) itself do not define any direction, they define quantity. You "assign" direction to them by placing them on a line, but that does not mean they indicate direction.
If i write the numbers 5, 8 and 10 on the line, by your logic they would also indicate direction relative to each other. But do they actually? (Hint: No) This is no different than positive and negative.
The modern notions that separate "scalars" and "vectors" goes as follows:
Scalars are elements of fields. Examples of fields include the rational numbers, the real numbers, and the complex numbers. Scalars can be added and multiplied and divided.
Vectors are spaces over fields. These are basically just lists of elements of fields (You can get fancier than that, but let's not). Velocity vectors, for example, are triples of real numbers. Vectors can be added and subtracted, but not multiplied nor divided.
That's it! Here are some examples.
Real scalars that can have a minus sign include the coulomb and gravitational potentials, as well as any other potential (like $\mathrm{Pe}=mgh$, $h$ can be negative).
Complex numbers are scalars with no notion of positive or negative (you cannot say $i<0$ or $i>0$). They do have a notion of "direction", but in quantum mechanics for example, the "direction" of a complex number is meaningless (we say "the wavefunction is unique up to a phase"), so you really do have scalars with no possible meaning of "direction", but which still have $+1$ and $-1$ as scalars.
Temperature in kelvin, and mass, are both [real] scalars almost always positive. But it's incorrect to say "that's the case because mass and temperature are scalars". There are other reasons that's the case.
Middle school teachers might tell you that "$-3$ cannot be a scalar, because it has a direction", but my example with the gravitational potential is a good counter-argument, and my "wavefunction" example seals the deal. If your teacher insists "$-3$ cannot be a scalar", memorize their sentence, remember it on their test, and forget it immediately afterwards :)
Think of a vector as having direction in space (north, south, east, west). Scalars may or may not be capable of having negative values. It just depends on the nature of the quantity.
The statement that negative values for scalars are just convention is rather misleading. Some "conventions" just naturally make a whole lot of sense, and changing them would be illogical. (and btw, despite another answer, it is possible to have negative temperature, even with the Kelvin scale, or with any scale. Just look at any college level text on thermodynamics.)
A negative value for a scalar does not imply a direction in space. Temperature can be negative, but temperature doesn't have a direction in space. A circuit can have a negative amount of voltage at a given point, but the voltage isn't pointing in any direction. Work can be negative, meaning you're taking energy from something. (think of friction slowing a car down.)
The "one-dimensional axis" that you're referring to isn't something in real space. Does this axis point towards the north, the south, the east, or the west? Or does it point into the sky? It's just a concept that you (or your teacher, or someone) has invented to help you understand things.