Inclusion Criteria:

• Subject completed the SHP621-302 (NCT02736409) extension study and is considered by the investigator to potentially benefit from continued BOS investigational treatment.
• Subject is able to provide written informed consent (subject, parent or legal guardian and, as appropriate, subject assent) to participate in the study before completing any study-related procedures.
• Females of childbearing potential must agree to continue acceptable birth control measures (example (eg): abstinence, surgically sterile male partner, stable oral contraceptives, or double-barrier methods) throughout study participation.
• Subject is willing and has an understanding and ability to fully comply with study procedures and restrictions as defined in protocol.

Exclusion Criteria:

• Subject has changes in medications or diet during the SHP621-302 (NCT02736409) study that could affect participation in this continuation study.
• Subject anticipates using swallowed topical corticosteroid for EoE or systemic corticosteroid for any condition during the treatment period; any temporary use (less than or equal to [≤] 7 days) or initiation of new steroid treatment during the study should be documented and discussed with the medical monitor prospectively but should be avoided within 4 weeks of the scheduled esophagogastroduodenoscopy (EGDs).
• Subject anticipates use of Cytochrome P450 3A4 inhibitors (eg, ketoconazole, grapefruit juice) during the continuation study.
• Subject has an appearance at the EGD at the final treatment evaluation visit of SHP621-302 (NCT02736409) (Visit 8) of an esophageal stricture (high grade), as defined by the presence of a lesion that does not allow passage of a diagnostic adult upper endoscope (eg, with an insertion tube diameter of greater than (>) 9 millimeter [mm]).
• Subject has presence of esophageal varices at the EGD at the final treatment evaluation visit (Visit 8) of the SHP621-302 (NCT02736409) study.
• Subject has any current disease of the gastrointestinal tract, aside from EoE, including eosinophilic gastritis, enteritis, colitis, or proctitis, inflammatory bowel disease, or celiac disease.
• Subject has other diseases causing or associated with esophageal eosinophilia, including hypereosinophilic syndrome, collagen vascular disease, vasculitis, achalasia, or parasitic infection.
• Subject has oropharyngeal or esophageal candidiasis that failed to respond to previous treatment. Diagnosis with oropharyngeal or esophageal candidiasis at or since the final treatment evaluation visit (Visit 8) of the SHP621-302 (NCT02736409) study is not an exclusion as long as the subject is expected to respond to treatment.
• Subject has a potentially serious acute or chronic infection or immunodeficiency condition, including tuberculosis, fungal, bacterial, viral/parasite infection, ocular herpes simplex, or chicken pox/measles.
• Subject has upper gastrointestinal bleeding identified at the EGD at the final treatment evaluation visit (Visit 8) of the SHP621-302 (NCT02736409) study.
• Subject has evidence of active infection with Helicobacter pylori.
• Subject has evidence of unstable asthma.
• Subject is female and pregnant or nursing.
• Subject has a history of intolerance, hypersensitivity, or idiosyncratic reaction to budesonide (or any other corticosteroids), or to any other ingredients of the study medication.
• Subject has a history or high risk of noncompliance with treatment or regular clinic visits