How about

Histogram[poisson = RandomVariate[PoissonDistribution[1], 1000000], {-1/2, Max[poisson] + 1/2, 1}]

I think there are two issues: (1) How to get done what you want to get done?, and (2) Should you be doing what you want?

To answer the second question one would need to know the purpose for your graphic. For the first part another option is to use the BarChart function:

SeedRandom[12345];
poisson = RandomVariate[PoissonDistribution[1], 500];
BarChart[Last[HistogramList[poisson]], BarSpacing -> 3,
 ChartLabels -> Range[Min[poisson], Max[poisson]]]

It finally dawned on me: I think you want the binwidth parameter to be the width of the bar as opposed to a width of a "bin" that might encompass more than one value.

I know these forums are typically "how to do something" rather than "should I do this?" but I think for graphics (and statistical analyses) it's critical to know what is the kind of message to the viewer that's desired. For example, I'm just not seeing the need for separating the bars or even to have bars. Here is a DiscretePlot:

DiscretePlot[Length[poisson] PDF[EmpiricalDistribution[poisson], s],
 {s, 0, Max[poisson]}, AxesOrigin -> {-0.5, 0}]

If something looks like a Graphics display, it very well might be, and if you look at the FullForm of your histogram, you'll see that the output is indeed a Graphics "object". That suggests that Histogram will accept the same options that Graphics does. There might be overrides or other imperfect matches, but basically you can expect to have the same control over the output that you would for Graphics. And the documentation for Histogram says this explicitly:

Histogram has the same options as Graphics with the following additions and changes: ...

One option that Graphics accepts is Ticks, and if I'm understanding your question, that's the relevant one for you. Look up the documentation for Ticks to get the specifics, but here's an example:

With[
 {poisson = RandomVariate[PoissonDistribution[1], 100]},
 Histogram[poisson, {1/2}, Ticks -> {Range[Min[poisson] + 1/4, Max[poisson] + 1/4, 1], Automatic}]]