Can I write my signature in my conlang's (constructed language) script, which consists entirely of straight lines? If not, why?
A legally binding signature doesn't have to be written in cursive, or in English, or even be your name, or any text at all for that matter. The purpose of a signature is to identify the person who is signing the document, and to indicate that they have understood and approved the document. So long as you have a reasonably unique and reproducible mark, your signature can be pretty much anything, from your name, to a pseudonym, to pictograph. A classic trope in media is for pirates or peasants or other illiterate people to simply sign contracts with an "X" or to "make their mark", since they are unable to write their own name.
Now, using things other than your name written in cursive might be frowned upon and difficult to implement in practice, since a pictograph can't be as easily linked to an individual as a name can, and signing a contract with something other than a recognizable form of your name is a downright unusual way to conduct business. When signing for a mortgage, for example, a lender might get fears that you're trying to defraud them if you start signing all your documents with a smiley face. But in principle, signing a mortgage with a smiley face or conlang script is exactly as binding as signing your real name in cursive.
Local laws may vary in what constitutes a signature, but it seems this is generally the case.
The laws on what constitutes a valid signature vary in different jurisdictions. But in many any sign intended to represent the signer's assent, and generally recognized as having such a meaning, is a legally valid signature. A typed name or initials at the end of an email is now a legal signature in many jurisdictions for many purposes. But if the mark is not known to others as being used to represent your signature, they may not accept its use as indicting an intent to accept the document. Using a more usual form of signature is less likely to cause hassle and contention.