Making square brackets more similar to \lfloor

\documentclass{article}
\usepackage{amssymb,stackengine}
\newcommand\lbr{%
  \ensurestackMath{\stackengine{0pt}{\lfloor}{\lceil}{O}{c}{F}{F}{L}}}
\newcommand\rbr{%
  \ensurestackMath{\stackengine{0pt}{\rfloor}{\rceil}{O}{c}{F}{F}{L}}}
\begin{document}
\begin{equation}
\lbr a \rbr
\end{equation}
\end{document}

Those are font decisions so you should switch to another font.

Cambria Math, TeX Gyre Pagella Math, DejaVu Math TeX Gyre and Neo Euler might do the trick.

If you want to stick to your current font, you will have to construct your own symbol by soperposing lfloor and lceil. You will find many examples of such things on this homepage. Personally, I would not touch the spacing of square brackets but I leave this to you.

The appearance of your brackets eventually depends on your fonts, so you might want to look at some different math fonts. That said, you can superimpose \lfloor and \lceil without any additional packages using \ooalign:

\documentclass{article}

\usepackage{amsmath}

\renewcommand\lbrack[1][]{\ooalign{\ensuremath{#1\lfloor}\cr\ensuremath{#1\lceil}}}
\renewcommand\rbrack[1][]{\ooalign{\ensuremath{#1\rfloor}\cr\ensuremath{#1\rceil}}}

\begin{document}

\(\lbrack x \rbrack\)

\end{document}

I added an optional argument, so you can scale the commands, e.g. \lbrack[\Big]. Note that you cannot use \left or \right with these commands.

That said, I would recommend not to alter the appearance of brackets like this and rather look for fonts that you like better.