"Dropping Out" after one semester would likely result in ADSEP and NOT Discharge. To obtain VA benefits you need either a DD-214 showing time and character of service, or become Disabled to a compensable degree while on Active Duty as determined by a MEB and PEB, or obtain a "Hardship" discharge.
§ 3.12a Minimum active-duty service requirement.
(a) Definitions.
(1) The term minimum period of active duty means, for the purposes of this section, the shorter of the following periods.
(i)
Twenty-four months of continuous active duty. Non-duty periods that are excludable in determining the Department of Veterans Affairs benefit entitlement (e.g., see § 3.15) are not considered as a break in service for continuity purposes but are to be subtracted from total time served.
(ii) The full period for which a person was called or ordered to active duty.
(2) The term benefit includes a right or privilege but does not include a refund of a participant's contributions under 38 U.S.C. Ch. 32.
(b) Effect on Department of Veterans Affairs benefits. Except as provided in
paragraph (d) of this section,
a person listed in paragraph (c) of this section who does not complete a minimum period of active duty is not eligible for any benefit under title 38, United States Code or under any law administered by the Department of Veterans Affairs based on that period of active service.
Para d exempts those individuals who are released on a "Hardship" or Disability" discharge and issued a DD-214. Para c defines inclusive those persons who may be eligible for VA benefits (24 months, or delayed entry and 24 months of AD)
As
@Capt MJ stated, every State has differing inclusions, requirements, and benefits, so while not eligible for Federal VA benefits, they very may be eligible for various State administered benefits.