If I use TeX Live 2015, the output is correct, so the problem seems to belong in fvextra.sty
(a recent addition). Indeed the package code has
unprotected end-of-lines:
1370 \def\FV@VerbatimPygments#1#2{%
1371 \edef\FV@PYG@Literal{\expandafter\FV@DetokMacro@StripSpace\detokenize{#1}}%
1372 \def\FV@BreakBeforePrep@PygmentsHook{%
1373 \expandafter\FV@BreakBeforePrep@Pygments\expandafter{\FV@PYG@Literal}}
1374 \def\FV@BreakAfterPrep@PygmentsHook{%
1375 \expandafter\FV@BreakAfterPrep@Pygments\expandafter{\FV@PYG@Literal}}
1376 \ifx#2\relax
1377 \let\FV@PYG#1
1378 \else
1379 \let\FV@PYG#2
1380 \fi
1381 \ifbool{FV@breakbytoken}%
1382 {\ifbool{FV@breakbytokenanywhere}%
1383 {\def\FV@BreakByTokenAnywhereHook{%
1384 \def\FV@BreakByTokenAnywhereBreak{%
1385 \let\FV@BreakByTokenAnywhereBreak\FancyVerbBreakByTokenAnywhereBreak}}%
1386 \def#1##1##2{%
1387 \FV@BreakByTokenAnywhereBreak
1388 \leavevmode\hbox{\FV@PYG{##1}{##2}}}}%
1389 {\def#1##1##2{%
1390 \leavevmode\hbox{\FV@PYG{##1}{##2}}}}}%
1391 {\def#1##1##2{%
1392 \FV@PYG{##1}{\FancyVerbBreakStart##2\FancyVerbBreakStop}}}%
1393 }
(line numbers added for reference).
If I add %
at the end of lines 1373, 1375, 1377 and 1379, the output is correct:
\documentclass{scrbook}
\usepackage{minted}
\setmintedinline{breaklines=true}
\makeatletter
\def\FV@VerbatimPygments#1#2{%
\edef\FV@PYG@Literal{\expandafter\FV@DetokMacro@StripSpace\detokenize{#1}}%
\def\FV@BreakBeforePrep@PygmentsHook{%
\expandafter\FV@BreakBeforePrep@Pygments\expandafter{\FV@PYG@Literal}}% <---
\def\FV@BreakAfterPrep@PygmentsHook{%
\expandafter\FV@BreakAfterPrep@Pygments\expandafter{\FV@PYG@Literal}}% <---
\ifx#2\relax
\let\FV@PYG#1% <---
\else
\let\FV@PYG#2% <---
\fi
\ifbool{FV@breakbytoken}%
{\ifbool{FV@breakbytokenanywhere}%
{\def\FV@BreakByTokenAnywhereHook{%
\def\FV@BreakByTokenAnywhereBreak{%
\let\FV@BreakByTokenAnywhereBreak\FancyVerbBreakByTokenAnywhereBreak}}%
\def#1##1##2{%
\FV@BreakByTokenAnywhereBreak
\leavevmode\hbox{\FV@PYG{##1}{##2}}}}%
{\def#1##1##2{%
\leavevmode\hbox{\FV@PYG{##1}{##2}}}}}%
{\def#1##1##2{%
\FV@PYG{##1}{\FancyVerbBreakStart##2\FancyVerbBreakStop}}}%
}
\makeatother
\begin{document}
A sufficently long text, to make breaking meaningful.
Lets have some really cool code now \mintinline{text}{cool code}.
Some more code \mintinline{sql}{Test} an how about some Java \mintinline{java}{String a = "hello";}
\end{document}

Update
With version 1.2.1 of fvextra
(released 2016/09/02), the issue has been fixed, so the extra code is no longer necessary.