Hello Mark and All,
I do not want to be in disagreement with George but there are flaws  to the
feed,feed feed approach. Simply put you can get in trouble in the Midwest
with George's approach. Each area  IS different. Starting the first of April
and feeding till fall even with a package  will get  comb drawn but will
also get a swarm  if only kept in two boxes. We have got plenty of pollen
(unlike many areas of the U.S.)and bees being fed   raise plenty of brood.
Young queens swarm also when conditions are right.
 I proved this very fact to a friend once by shaking more and more bees in a
single box with a  young queen. Three hours later the young egg laying queen
swarmed exactly like I said she would.
Mark wrote:
 However, I will be traveling in June and will not be able to feed them for
a couple of weeks. Should I continue to give them sugar syrup knowing it
will run out?

You have been feeding for eight weeks. Check the bees progress. Rain will be
the only reason you might have to feed in June. If dry I would not feed as
June is our major honey flow.. Start back feeding in July if foundation is
not yet drawn or the hive feels light when tilted and  sweet clovers  have
quit blooming.

>Should I make a big enough piece of candy to last the time period I'm gone?

No. Bees prefer nectar over candy and  syrup once they get strong enough and
your bees should be approaching 25 to 30,000 plus by now.

What about dry sugar on the inner cover? Is that a possibility?

No!  Dry sugar is for starvation winter feeding for those beekeepers which
do not properly prepare their hives for winter. I never use dry sugar but
will work if your bees are starving.

< I'm in SE Missouri and in June there should presumably be a fairly good
nectar flow.

We get years without the state average in honey production but all we need
this year is for the rain to stop (like they are predicting for next week as
the jet stream is finnally changing) to get at least a average honey
production year. The plants   are   ready to bloom. I can not predict the
start of the main flow for your area without   looking  at your location.
The time frame varies but your main flow should be a full week ahead of my
area. The late date for the main flow is June  10-15 in my area and can
start much earlier. Flows can start and stop in Missouri with rain and  cool
weather. I have only scratched the surface on Missouri plant discussion but
giving a few helpful hints. Learn the honey plants in your area.

Floral sources can vary  every few miles in Missouri.  The worst location is
in the middle  of farmers fields for as far as you can see.

 Black Locust is in bloom now but duration is short but flows can be intense
if the bees are strong.  The bees did not produce black Locust honey last
year even though the trees were white with blooms. I do not have a clue as
to the reason.
possible reasons:
cool nights
cool days
subsoil moisture
humidity
rain

 Dutch Clover is blooming but the rain will have to stop before the flow can
start. I saw the bees on the  Dutch Clover for about three hours  last week
until the rain came. Even Dan Rather is talking about our rainy weather.

We had good minor flows from henbit and  wild mustard this spring in most
areas.

Good locations will make a successful beekeeper and poor choice locations
have been the ruin of many a beekeeper. Once I choose a location I leave the
bees in the location for three years before evaluating the location.

Sincerely,
Bob Harrison
Odessa, Missouri

Ps. Taking extra time with the new Missouri beekeeper.