Alignment tab character inside a starred command within align

TL;DR

Simple solution: use a more robust check for the *-variant.

\documentclass{article}

\usepackage{amsmath}

\NewDocumentCommand{\mylinebreak}{s}{%
  \IfBooleanTF{#1}{&x\\}{&x&=y\\}%
}

\begin{document}
\begin{align*}
 a&=b\mylinebreak
  &=c.\\
 a&=b\mylinebreak*
  &=c.\\
\end{align*}
\end{document}

I added something in order to better see that the choices are made correctly.

If you're running a version of LaTeX prior to the 2020-10-01 release, you also need \usepackage{xparse}.

Analysis of the issue

Let's see what happens: \mylinebreak becomes

\@ifstar\@mylinebreak\@@mylinebreak

According to the definition of \@ifstar by amsmath

\def\@ifstar#1#2{\new@ifnextchar *{\def\reserved@a*{#1}\reserved@a}{#2}}

the above becomes

\new@ifnextchar*{\def\reserved@a{\@mylinebreak}\reserved@a}{\@@mylinebreak}

Let's see what \new@ifnextchar does:

\long\def\new@ifnextchar#1#2#3{%
  \let\reserved@d= #1%
  \def\reserved@a{#2}\def\reserved@b{#3}%
  \futurelet\@let@token\new@ifnch
}

so the expansions continues as (newlines just for reading convenience)

\let\reserved@d= *
\def\reserved@a{\def\reserved@a*{\@mylinebreak}\reserved@a}
\def\reserved@b{\@@mylinebreak}
\futurelet\@let@token\new@ifnch &

The trailing & comes from the fact that the newline after \mylinebreak is ignored during tokenization. Unfortunately, we're in an alignment, so as soon as TeX scans &, it inserts the v part of the template. And, indeed, the error message with a high value of \errorcontextlines reads

! Misplaced alignment tab character &.
\@@mylinebreak ->&
                  &\\
<to be read again> 
                   }
<template> }
            $}\ifmeasuring@ \savefieldlength@ \fi \set@field \hfil \endtempl...
<argument>  a&=b\mylinebreak &
                              =c. 

and \@let@token is set to }. This is confirmed by injecting \show\@let@token at the beginning of \new@ifnch:

\@let@token=end-group character }

OK, let's go a step forward: the expansion of \new@ifnch yields

\ifx@let@token\reserved@d\let\reserved@b\reserved@a\fi\reserved@b

In our case, the \ifx test returns false, so we remain with

\reserved@b}&

and this breaks, because \reserved@b becomes \@@mylinebreak.

You are defining \\ to look ahead for a * (incidentally the standard command already looks ahead for a *) but in doing this it "sees" the & and the next cell starts before the previous row is ended by inserting the original \\

The definition in amsmath is

\def\math@cr{\relax\iffalse{\fi\ifnum0=`}\fi
  \@ifstar{\global\@eqpen\@M\math@cr@}%
          {\global\@eqpen
             \ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
              \else -\@getpen\dspbrk@lvl \fi
           \math@cr@}}
\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
\def\math@cr@@[#1]{\ifnum0=`{\fi \iffalse}\fi\math@cr@@@
  \noalign{\vskip#1\relax}}

The special \relax\iffalse{\fi\ifnum0=`}\fi is designed to avoid this issue by preventing the halign template ending a cell at this point.

Note that it is already using \@ifstar to define \\* to prevent a page break at this point.

It is not clear what is the intention of using &\\ as the behaviour of that construct is the same as \\ as the empty cells do not do anything in almost all circumstances.