How can I implement a binary chirp signal containing zeros and ones?
What is a chirp signal?

A chirp signal is a sinusoidal signal of the form $\sin(bt+\frac{1}{2}at^2)$ whose instantaneous radian frequency (the derivative with respect to time of the argument of the sine, viz. $b+ at$, is a linear function of $t$. If the chirp signal is of
duration $T$, then the instantaneous frequency at the end is $B=b+aT$.
If $a &gt; 0$, the frequency increases
with time from the initial value $b$ to its maximum value $B$ at $t = T$,
while if $a&lt;0$, the frequency decreases from
$b$ downwards to the minimum value $B$ at $t = T$.

Is the Fourier transform of a chirp signal constant over 
the frequency range $[b,B]$  (or $[B,b]$ for pessimists who choose $a &lt; 0$)?

No, the Fourier transform is not constant over this
frequency range.  Nor does the DFT have equal amplitude
in all bins if we sample the chirp signal and compute its DFT.

What is a fake chirp signal?

A fake chirp signal is one whose instantaneous radian frequency
increases in steps of equal height $\Delta = (B-b)/n$ but decreasing 
durations so that it consists of one full period of $\sin(bt)$ 
followed by one full period of $\sin((b+\Delta)t)$ followed by one 
full period of $\sin((b+2\Delta)t)$, $\ldots$, followed by one full period
of $\sin((b+(n-1)\Delta)t) = \sin((B-\Delta)t)$.  Note that the total
duration is
$$T = 2\pi\left[\frac{1}{b} + \frac{1}{b+\Delta} 
+ \cdots + \frac{1}{b+(n-1)\Delta}\right]$$
To maintain phase continuity, we insist that each full period start 
and end at $0$ amplitude and $0$ phase so that, for example, between 
$t=\frac{2\pi}{b}$ and $t=\frac{2\pi}{b}+\frac{2\pi}{b+\Delta}$ when
we have a sinusoid of radian frequency $b+\Delta$, 
the exact mathematical expression
for the signal is $\sin((b+\Delta)t+\theta)$ where $\theta$ is chosen
so that $(b+\Delta)\frac{2\pi}{b}+\theta$ is an integer multiple of $2\pi$.

What is a binary chirp signal?

If you create a chirp signal, sample it, and hard-limit it to $\pm 1$ 
(or, as you prefer, shift the dc level so that the signal is $0$s and $1$s) 
then you can call it a binary chirp signal, but you shouldn't
expect it to have the properties one expects of a chirp
signal.  In any case, it is necessary to have $b,B \ll f_s$
where $f_s$ is the sampling rate to get something reasonable.
If you don't want to go through the hassle of creating a sampled
chirp signal, etc. you can get something that can be called
a faker  binary chirp signal very simply.  Here are
two examples.

$$111110000011110000111000110010~\text{or}~1111100001110010$$

The first sequence is effectively one period of a sinusoid with $10$ samples
per period followed by one period of a sinusoid with $8$ samples
per period followed by one period of a sinusoid with $6$ samples per period,
etc.  The second sequence has half periods of sinusoids with decreasing
numbers of samples per period.  I call this a faker binary chirp
signal because instead of the frequency of the sinusoid
increasing in constant steps, it is the period of the sinusoid
that is decreasing in constant steps.  So we don't have the linear
frequency sweep that characterizes chirp signals.  But the signal is
trivial to synthesize, and if $b, B \ll f_s$, the difference between
a fake binary chirp and a faker binary chirp can be small.

I don't read MATLAB but if x = 1:0.1:1e3; means that it is desired 
that the frequency sweep from $1$ Hz to $1000$ Hz (or by a factor of $1000$
in general), then the digital sequence (whether sampled from a true chirp
pr a fake chirp or synthesized as a faker chirp) is going to begin with 
long strings of $1$s followed by long strings of $0$s.  Matters 
will be even worse
if a frequency resolution or step size $\Delta$ of $0.1$ Hz is required.