You could use CustomJS to insert some JavaScript code that will be used to change the bottom and top values of the BoxAnnotation. I'm using the Slider from Bokeh in this example:
from bokeh.io import vform
from bokeh.models import CustomJS, Slider
from bokeh.plotting import figure, show
from bokeh.models import BoxAnnotation
plot = figure(plot_width=300, plot_height=300)
plot.line([0,1],[0,1], line_width=3, line_alpha=0.6)
box_l = BoxAnnotation(plot=plot, top=0.4,
fill_alpha=0.1, fill_color='red')
box_m = BoxAnnotation(plot=plot, bottom = 0.4,top=0.6,
fill_alpha=0.1, fill_color='green')
box_h = BoxAnnotation(plot=plot, bottom=0.6,
fill_alpha=0.1, fill_color='red')
plot.renderers.extend([box_l, box_m, box_h])
callb_low = CustomJS(args=dict(box_l=box_l,box_m=box_m,plot=plot),
code="""
var level = cb_obj.get('value')
box_l.set({"top":level})
box_m.set({"bottom":level})
plot.trigger('change');
""")
callb_high = CustomJS(args=dict(box_m=box_m,box_h=box_h,plot=plot),
code="""
var level = cb_obj.get('value')
box_m.set({"top":level})
box_h.set({"bottom":level})
plot.trigger('change');
""")
slider1 = Slider(start=0.1, end=1, value=0.4, step=.01, title="low",
callback=callb_low)
slider2 = Slider(start=0.1, end=1, value=0.6, step=.01, title="high",
callback=callb_high)
layout = vform(slider1,slider2, plot)
show(layout)
The output will look like:

Based on the suggestions by bigreddot, you can have the following done in ipython notebooks:
from bokeh.io import push_notebook
from bokeh.plotting import figure, show
from bokeh.models import BoxAnnotation
from ipywidgets import interact
p = figure(x_range=(0,1), y_range=(0,1),plot_width=300, plot_height=300)
box_L = BoxAnnotation(plot=p, top=0.4,
fill_alpha=0.1, fill_color='red')
box_M = BoxAnnotation(plot=p, bottom = 0.4,top=0.6,
fill_alpha=0.1, fill_color='green')
box_H = BoxAnnotation(plot=p, bottom=0.6,
fill_alpha=0.1, fill_color='red')
p.renderers.extend([box_L, box_M, box_H])
def update_func(thresh_L=0.4, thresh_H=0.6):
box_L.top = box_M.bottom = thresh_L;
box_M.top = box_H.bottom=thresh_H;
p.renderers.extend([box_L, box_M, box_H])
push_notebook() # note, just a function, not a method on "source"
show(p)
Then in a separate cell you start your sliders like:
interact(update_func, thresh_L=(0, 1, 0.1),thresh_H=(0, 1, 0.1))
